Preheat your oven to 350 degrees F (175 degrees C). Cut kielbasa into 3 to 4 large sections, then each section in half lengthwise. Slice each section into half moons about a half-inch long.
Combine brown sugar, mustard, vinegar, and pepper in a medium saucepan. Set over low heat and cook, stirring, until the sugar has dissolved completely. Stir in kielbasa.
Line a sheet pan with parchment paper. With a slotted spoon, transfer the kielbasa pieces from the pot and spread evenly over the parchment paper. Set aside leftover sauce in the pot.
Bake for 15 minutes.
Remove from oven, then carefully lift up the parchment paper and funnel kielbasa pieces back into the pot of sauce. Stir to coat again, then repeat the process of putting the kielbasa pieces back on the parchment-lined sheet pan. Bake for another 15 minutes.
Transfer to a serving platter and serve with toothpicks.